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1: 1.4 Calculus of One Variable
If f ( x 1 ) f ( x 2 ) for every pair x 1 , x 2 in an interval I such that x 1 < x 2 , then f ( x ) is nondecreasing on I . … If also f ( x ) is continuous on the right at x = a , and continuous on the left at x = b , then f ( x ) is continuous on the interval [ a , b ] , and we write f ( x ) C [ a , b ] . … If f ( x ) is continuous on an interval I save for a finite number of simple discontinuities, then f ( x ) is piecewise (or sectionally) continuous on I . … If f ( n ) exists and is continuous on an interval I , then we write f C n ( I ) . …
2: 26.2 Basic Definitions
If, for example, a permutation of the integers 1 through 6 is denoted by 256413 , then the cycles are ( 1 , 2 , 5 ) , ( 3 , 6 ) , and ( 4 ) . … Unless otherwise specified, it consists of horizontal segments corresponding to the vector ( 1 , 0 ) and vertical segments corresponding to the vector ( 0 , 1 ) . For an example see Figure 26.9.2. … A partition of a nonnegative integer n is an unordered collection of positive integers whose sum is n . As an example, { 1 , 1 , 1 , 2 , 4 , 4 } is a partition of 13. …
3: 28.17 Stability as x ±
However, if ν 0 , then ( a , q ) always comprises an unstable pair. …
4: 18.36 Miscellaneous Polynomials
These are OP’s on the interval ( - 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at - 1 and 1 to the weight function for the Jacobi polynomials. …
5: Bibliography K
  • R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.
  • 6: 18.2 General Orthogonal Polynomials
    It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … More generally than (18.2.1)–(18.2.3), w ( x ) d x may be replaced in (18.2.1) by a positive measure d α ( x ) , where α ( x ) is a bounded nondecreasing function on the closure of ( a , b ) with an infinite number of points of increase, and such that 0 < a b x 2 n d α ( x ) < for all n . … All n zeros of an OP p n ( x ) are simple, and they are located in the interval of orthogonality ( a , b ) . …
    7: 10.23 Sums
    provided that f ( t ) is of bounded variation (§1.4(v)) on an interval [ a , b ] with 0 < a < x < b < 1 . …
    8: 1.6 Vectors and Vector-Valued Functions
    c ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) , with t ranging over an interval and x ( t ) , y ( t ) , z ( t ) differentiable, defines a path. … with ( u , v ) D , an open set in the plane. …
    9: 2.3 Integrals of a Real Variable
    k ( ) and λ are positive constants, α is a variable parameter in an interval α 1 α α 2 with α 1 0 and 0 < α 2 k , and x is a large positive parameter. …
    10: 18.1 Notation
    x , y real variables.
    w ( x ) weight function ( 0 ) on an open interval ( a , b ) .